Today let’s take a break
from economic issues and get back to option pricing. In continuation with my
last piece on option pricing (Link) , wherein we explored a much easier way to price
options and what was even better was that it allowed for the flexibility to
have non-normal probability distribution in pricing. Today I would like to
write about how we can develop the concept further to include in the money and
out of money options. Well if one understands the basic premise of my last
article on this topic it’s fairly simple.

As I wrote before, basically if I
buy an “At the money” call and put, the price that I should pay is the expected
movement of the stock in that period. So in essence

C+P = E[S] –
Eq 1

Now for any
option that is “In the money” for the buyer to make any money and for the
seller to loose money would be when the Stock crosses the Strike price of the
option, so in essence what we really care about are the probabilities wherein
the Stock Price crosses the Strike Price of the option.

So the price
of the option would be:

C = (S-K) +E[ ∑

_{x=k}^{n}p_{x}*S]
The above formula gives the
flexibility to the reader to assume different probability distributions and use
them in the option pricing.

Please Note that If I try
to simplify the Black Scholes model i.e. take the assumption of Normal
probability distribution I would be only be approximating the pricing, but
nevertheless let me put it down anyways.

The call price (assuming
interest rates as negligible) c = SN(d1) – KN(d2), the put formula is also
accordingly KN(-d2) – SN(-d1),

c= SN(ln(S/K) + σ/2*√t) –
KN(ln(S/K) - σ/2*√t), Since We are taking volatility for the entire duration of
the option so I am going to replace σ*√t with σ .

Also I would express S in
terms of K and σ as that’s what we are concerned about, i.e.

S = K+ K*σ*Ɵ

Putting these values into
the Black Scholes formula and expanding the log portion using taylor equation
the log portion would be reduced to Ɵ – Ɵ^2* σ/2

Finally adding both these
terms and using the Normal expansion and exponential expansion by taylor series

N(x) =[∫

_{-∞}^{0}exp^^{- (x)^2 }+ ∫_{0}^{ σ/2}exp^^{- (x)^2}]/sqrt(2 π)
The first part is equal to
0.5, to integrate the second part of the expression let me expand it using the
Taylor series. The Taylor series expansion of the exponential term is

e

^{x}= 1 + x + x^2/2! + x^3/3! ……
Replacing x by Ɵ – Ɵ^2* σ/2+σ/2
and then integrating we get

N(Ɵ – Ɵ^2* σ/2+σ/2) = 0.5+
[

_{0}^{ σ/2}∑_{0}^{n}(-1)^{ n}(Ɵ – Ɵ^2* σ/2+σ/2 )^{2n+1}/2^{n}n!(2n+1)] /sqrt(2 π)
As SD is small number (in
decimals) we can ignore the higher power of σ,

Please note that if σ is
high the whole notion of Normality falls apart and as does the Black Scholes
model.

So expanding this
expression and putting it in Eq 1 we get

C = (2* Ɵ - Ɵ^2* σ/2 + σ) *
√(1/2π)

Or more simply C = (2* Ɵ + σ)
* √(1/2π)

Again this is simplying
Black Scholes using mathematical tools what’s more important to understand is
that the genesis of any option pricing i.e. using any probability distribution
is this formula below:

C = (S-K) +E[ ∑

_{x=k}^{n}p_{x}*S]
Till Next Time………….

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